Simple Pole Research Articles

Orthogonal polynomials with discontinuous weights

In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w_0 and a standard jump function, then A_n(z) and B_n(z) have apparent simple poles at these jumps. We exemplify the approach by taking w_0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painleve IV.

Approximation by the Riemann zeta-function

Any meromorphic function having at most simple poles can be approximated by linear combinations of translates of the Riemann zeta-function. In particular, an arbitrary holomorphic function can be so approximated. If derivatives of the zeta-function are allowed, then arbitrary meromorphic functions can be approximated.

Distributional Wiener-Ikehara theorem and twin primes

The Wiener-Ikehara theorem was devised to obtain a simple proof of the prime number theorem. It usesno other information about the zeta function ~(z) than that it is zero-free and analytic for Re z ⩾ 1, apart from a simple pole at z = 1 with residue 1. In the Wiener-Ikehara theorem, the boundary behavior of a Laplace transform in the complex plane plays a crucial role. Subtracting the principal singularity, a first order pole, the classical theorem requires uniform convergence to a boundary function on every finite interval. Here it is shown that local pseudofunction boundary behavior, which allows mild singularities, is necessary and sufficient for the desired asymptotic relation. It follows that the twin-prime conjecture is equivalent to pseudofunction boundary behavior of a certain analytic function.

Convolution Dirichlet Series and a Kronecker Limit Formula for Second-Order Eisenstein Series

AbstractIn this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Γ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Γ\H has finite volume yet is non-compact. Associated to each cusp of Γ\H, there is a classically studiedfirst-ordernon-holomorphic Eisenstein seriesE(s, z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein seriesE(s, z) admits a meromorphic continuation with a simple pole ats= 1. Classically, Kronecker’s limit formula is the study of the constant term1(z) in the Laurent expansion ofE(s, z) ats= 1. A number of authors recently have studied what is known as thesecond-orderEisenstein seriesE*(s, z), which is formed by twisting the Dirichlet series that defines the seriesE(s, z) by periods of a given cusp formf. In the work we present here, we study an analogue of Kronecker’s limit formula in the setting of the second-order Eisenstein seriesE* (s, z), meaning we determine the constant term2(z) in the Laurent expansion ofE*(s, z) at its first pole, which is also ats= 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function1. We then define two families of convolution Dirichlet series, denoted byandwithm∈ ℕ, which are formed by using the Fourier coefficients of1and the weight two cusp formf. We prove that for allm,andadmit a meromorphic continuation and are holomorphic ats= 1. Turning our attention to the second-order Kronecker limit function2, we first express2as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp formf, the Fourier coefficients of the first-order Kronecker limit function1, and special values(1) and(1) of the convolution Dirichlet series. Finally, we prove a bound for the special values(1) and(1) which then implies a bound for the Fourier coefficients of2. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.

On Designing Self-Tuning Controllers for AQM Routers Supporting TCP Flows Based on Pole Placement

This paper revisits the simple pole placement technique in the classical control theory, and exploits this technique to propose two kinds of controllers for active queue management (AQM) in Internet protocol (IP) routers: the self-tuning proportional controller based on pole placement (ST/spl I.bar/P/spl I.bar/PP) and the self-tuning proportional-plus-integral controller based on pole placement (ST/spl I.bar/PI/spl I.bar/PP). The damping ratio /spl xi/ and undamped natural frequency /spl omega//sub n/ can be appropriately chosen such that: 1) the transient response performance of the system is satisfied and 2) all the poles would lie in the left-half s-plane to guarantee the stability of the control system. The self-tuning controllers can assign proper intervals of /spl xi/ and /spl omega//sub n/ to achieve good AQM performance and thereby adapting the system to significant load changes very well. Furthermore, the ST/spl I.bar/PI/spl I.bar/PP controller can regulate the packet drop probability based on the knowledge of the instantaneous queue size, and clamp the steady value of the queue length to a specified reference value. We verify the effectiveness of these two controllers via OPNET simulation. Our simulation results show the following: 1) choosing appropriate /spl xi/ and /spl omega//sub n/ can successfully satisfy the transient response of the system and 2) when the network load changes, the ST/spl I.bar/P/spl I.bar/PP controller and the ST/spl I.bar/PI/spl I.bar/PP controller exhibit extremely short settling time.

IRREGULAR ISOMONODROMIC DEFORMATIONS FOR GARNIER SYSTEMS AND OKAMOTO'S CANONICAL TRANSFORMATIONS

In this paper we describe the Garnier systems as isomonodromic deformation equations of a linear system with a simple pole at zero and a Poincar\'e rank two singularity at infinity. We discuss the extension of Okamoto's birational canonical transformations to the Garnier systems in more than one variable and to the Schlesinger systems.

Identification of simple poles via boundary measurements and an application of EIT**This work is partly supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation.

We consider the problem of identifying simple poles of a meromorphic function by means of the value of the function measured on a circle enclosing those poles. We propose an algorithm for this problem in a mathematically rigorous way with a stability estimate. Results of numerical testing are provided to show validity of the algorithm. We then apply the method to an electrical impedance tomography problem to detect diametrically small inclusions of disc shape via boundary measurements.

Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Pole, with an Application to Associated Legendre Functions

Second‐order linear ordinary differential equations with a large parameter u are examined. Asymptotic expansions involving modified Bessel functions are applicable for the case where the coefficient function of the large parameter has a simple pole. In this paper, we examine such equations in the complex plane, and convert the asymptotic expansions into uniformly convergent series, where u appears in an inverse factorial, rather than an inverse power. Under certain mild conditions, the region of convergence containing the simple pole is unbounded. The theory is applied to obtain exact connection formulas for general solutions of the equation, and also, in a special case, to obtain convergent expansions for associated Legendre functions of complex argument and large degree.

Reductions of integrable equations: dihedral group

We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group - a group of twisted automorphisms of the corresponding infinite dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2,C) Lax operators with simple poles and corresponding integrable equations. In the last section we give three examples of dihedral reductions for sl(N,C) Lax operators.

On the dressing method for the generalised Zakharov–Shabat system

The dressing procedure for the generalised Zakharov–Shabat system is well known for systems, related to sl( N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ‘dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.

Zeros of the Derivative of a Rational Function and Coinvariant Subspaces for the Shift Operator on the Bergman Space

If all n (n > 1) zeros of a rational function r with simple poles are in a half-plane, then the derivative of r has at least one zero in the same half-plane. This result is used to prove that the number of zeros of a linear combination of n Bergman kernels in the unit disk may range from 0 to 2n-3. Bibliography: 7 titles.

Multivariable Rankin-Selberg integrals for orthogonal groups

We show a relation between a certain period and the existence of a simple pole for the spin and standard partial L functions corresponding to a generic cuspidal representation defined on the group GSO8(A). We also relate these two conditions with the functorial lift from the exceptional group G2 to GSO8. The main new ingridient is a new multivariable Rankin-Selberg integral which represents the above two L functions.

Estimating depth and model type using the continuous wavelet transform of magnetic data

The continuous wavelet transform has been proposed recently for the interpretation of potential field anomalies. Using Poisson wavelets, which are equivalent to an upward continuation of the analytic signal, this technique allows one to estimate the depth of burial of homogeneous field sources and to determine the nature of the source in the form of a structural index. Moreau et al. (1999) accomplish this by successively testing the least‐squares misfit on a log–log plot of the wavelet transform amplitude versus the sum of the depth and the dilation (upward continuation height). We extend this methodology by analyzing the ratio of the Poisson wavelet coefficients of the first and second orders. For simple pole sources, this ratio at one dilation is enough to estimate the depth and index uniquely; but for extended sources of finite size, we must analyze the variation of the estimates with dilations. The technique gives good results on synthetic and field examples.

AQM controller design for IP routers supporting TCP flows based on pole placement

The simple pole placement technique in classical control theory is exploited to propose two kinds of controllers for active queue management (AQM) in IP routers: the proportional controller based on pole placement (P―PP controller) and the proportional-plus-integral controller based on pole placement (PI―PP controller). The damping ratio ξ and the undamped natural frequency ω n are appropriately chosen such that the transient response of the system is satisfied and all the poles lie in the left-half s-plane to guarantee the stability of the system. The authors verify the effectiveness of the controllers via extensive simulations using OPNET, including the case of different time-varying round-trip times (RTTs) for different connections. Simulation results show that choosing appropriate values of ξ and ω n can successfully satisfy the transient response of the system. The impact of different values of ξ and ω n on the system performance is also investigated.

Nonvanishing derivatives and normal families

We consider the differential operators Ψ k , defined by Ψ1(y) =y and Ψ k+1(y)=yΨ k y+d/dz(Ψ k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z 2+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ k (f ′/f) =f (k)/f, we deduce in particular that iff andf (k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ′/f :f ∈F} is normal.

Integral and derivative dispersion relations, analysis of the forward scattering data

Integral and derivative dispersion relations (DR) are considered for the forward scattering $pp$ and $\bar pp$ amplitudes. A new representation for the derivative DR, valid not only at high energy, is obtained. The data on the total cross sections for $pp (\bar pp)$ interaction as well as the data on the parameter $\rho $ are analyzed within the various forms of the DR and high-energy Regge models. It is shown that three models for the Pomeron, Simple pole Pomeron, Tripole Pomeron and Dipole Pomeron (the both with the intercept equal unit) lead to practically equivalent description of the data at $\sqrt{s}>$5 GeV. It is also shown that the correctly calculated low-energy part of the dispersion integral (from the two-proton threshold up to $\sqrt{s}=$5 GeV) allows to reproduce well the $\rho $ data at low energies without additional free parameters.

Tunneling dissociation from a double well via path integrals

It is shown how the semiclassical theory of path integrals can be implemented in a practical manner for the analysis of a potential that combines the two-state system of a double well potential (DWP) with decay into a continuous spectrum. This potential may correspond to a variety of physical situations in physics and chemistry. The structure of the formalism and of the results is such that it allows computation not only for analytic but also for numerically given potentials. The central theme is the determination of the energy-dependent Green’s function, which is shown to consist of a regular part and a part containing simple and double complex poles. These poles represent the position of the energy levels, as well as the energy widths and shifts due to the interaction with the continuous spectrum. When applied to the bound DWP without tunneling, the theory is shown to reduce in certain limits to known results from the Jeffreys–Wentzel–Kiamers–Bhrillouin approximation. If the system is taken to be prepared in the first well, the interactions with the remaining of the potential lead to two types of transition rates. One represents the transient motion toward a virtual equilibrium state of the DWP. It emerges as a positive imaginary part of the self-energy. The other represents the decay into the continuum and emerges as a negative imaginary part of the pole. Comparison of the two mechanisms of nonstationarity is made for different magnitudes of the second barrier relative to the first one. Since the system decays to the continuum while oscillating, the theory obtains a correction to the frequency of oscillation in the DWP due to the interaction with the continuum. This phenomenon is observable in real two-state systems, if an external perturbation which affects mainly one state converts it into a resonance state.

How to Simplify Differentiation When Performing Partial Fraction Expansion in the Presence of a Single Multiple Pole

A new method is suggested to handle partial fraction expansion in the presence of a single high order pole, by using polynomial division and partial fraction expansion with simple poles only, to simplify the process of repeated differentiation of a rational function.

General subtraction method for numerical calculation of one-loop QCD matrix elements

We present a subtraction scheme for eliminating the ultraviolet, soft, and collinear divergences in the numerical calculation of an arbitrary one-loop QCD amplitude with an arbitrary number of external legs. The subtractions consist of local counter terms in the space of the four-dimensional loop momentum. The ultraviolet subtraction terms reproduce MSbar renormalization. The key point in the method for the soft and collinear subtractions is that, although the subtraction terms are defined graph-by-graph and the matrix element is also calculated graph-by-graph, the sum over graphs of the integral of each the subtraction term can be evaluated analytically and provides the well known simple pole structure that arises from subtractions from real emission graphs, but with the opposite sign.

Closed Semistable Operators and Singular Differential Equations

We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which 0 is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of C0-semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly perturbed differential equation in a Banach space.